Finite Group Databasebeta

← Prev (#6303) Next (#6305) →

Cayley diagram

C3×C4×C3⋊D3 #6304

order216
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C23×AGL(2,𝔽3) (order 3456) · Inn(G) = C3⋊D3 · Out(G) = C23×S4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C4×C3⋊D3

Splittings

FittingC33×C4 ⋊ C2
derivedC32 ⋊ C2×C3×C4
SylowC33 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC3×C4C3⋊D3
commutator, hypocenterC32C2×C3×C4
FrattiniC2C2×C3×C3⋊D3
FittingC33×C4C2
socleC2×C33C22
prime radicalC2×C3×C3⋊D3C2
cosocleC2×C32C22×C3
LCSG ▷ C32
UCSC1 ◁ C3×C4

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 13 points:

(5,6,7)(9,10)(12,13);(1,4,3,2)(11,12,13);(8,9,10)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13165192, #13165193, #13165194, …
maximal subgroup ofC3×#3710, C3×#3712, C3⋊D3×C3⋊C8, …
maximal normal subgroup ofC3×#3710, C3×#3712, C3⋊D3×C3⋊C8, …
maximal quotient ofC3×C8×C3⋊D3, C3×#3743, C3×C4×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 141);